Theorem 2.3 (Theorem A, [1]}) 
Let \(f\)  be a homeomorphism of \(\mathbb {A}\)  which is isotopic to the identity and let \(\check{f}\)  be a lift of \(f\)  to \(\mathbb {R}^2\) . We suppose that \(f\)  has no topological horseshoe. Then each point \(z\)  such that its \(\omega \) -limit set is non empty has a well defined rotation number \(\operatorname{Rot}_{\check{f}}(z)\) , i.e. for every compact set \( K \subset \mathbb {A}\)  and every increasing sequence of integers \((n_k)_{k\in \mathbb {N}}\)  such that \(z\)  and \(f^{n_k}(z)\)  belong to \(K\) , we have
\( \lim _{k\rightarrow \infty } \frac{1}{n_k}(\pi _1(\check{f}^{n_k}(\check{z})-\check{z} ) )= \operatorname{Rot}_{\check{f}}(z) ,\) 
